CN105630608A - Method for achieving multiprocessor scheduling through combined cross entropy - Google Patents

Abstract

The invention discloses a method for achieving multiprocessor scheduling through combined cross entropy. The method comprises the following steps that 1, a mathematical model of the multiprocessor scheduling problem is built, and an objective function and constraint conditions are determined; 2, programming and solving are conducted on the determined objective function and constraint conditions for the multiprocessor scheduling problem through a combined cross entropy algorithm and Matlab, and an optimal solution of the objective function is obtained. According to the method for achieving multiprocessor scheduling through the combined cross entropy, on the basis of a constraint relationship between processors and operations, the multiprocessor scheduling problem is represented as a linear 0-1 integer programming problem to achieve minimality of the objective function, and the optimal solution of the problem is solved through the combined cross entropy algorithm. The combined cross entropy algorithm has the advantages that the optimal solution can be obtained, the number of iterations is small, the stability is high, and the running time is short; the effectiveness and practicability of the algorithm on handling the multiprocessor problem are further proved.

Description

Combined cross entropy is utilized to realize the method for multiprocessor scheduling

Technical field

The present invention relates to a kind of multiprocessor scheduling method, it is specifically related to a kind of method utilizing combined cross entropy to realize multiprocessor scheduling.

Background technology

The main trend of world today's development is parallelization, networking, intellectuality, and the calculating of parallel distributed formula is one of main bugbears of these development, is also one of hot issue working as the research of prescience neighborhood. The coordination of the design of parallel algorithm, the division of task, communication and scheduling synchronous, multitask are that current parallel distributed formula calculates the problem needing to solve, and the scheduling meeting of task directly affects the efficiency of calculating, so how rationally carrying out the scheduling of multitask efficiently and distribution is current urgent need to solve the problem. So the research of the application of Problem of Multiprocessor Scheduling and theoretical aspect just obtains scientific domain investigator and extremely pays close attention to. In actual life, the problem of a lot of planning tasks distribution is all closely related with Problem of Multiprocessor Scheduling. The problem with Combinatorial Optimization character such as fields such as engineering technology, Parallel and distributed computation, industrial and agricultural production and the communications and transportation of living closely bound up with us just can be converted into Problem of Multiprocessor Scheduling and solve. Problem of Multiprocessor Scheduling is substantially np complete problem, and traditional linear programming cannot solve, and current many employing heuritic approaches both at home and abroad are such as simulated annealing, ant group algorithm, quanta particle swarm optimization approximate solution. But mostly there is the problems such as speed of convergence is slow, efficiency is lower in these algorithms.

Cross-Entropy Algorithm (CrossEntropyAlgorithm, CE) it is a kind of overall situation random optimization algorithm proposed based on the cross entropy theory in information theory, this algorithm utilizes the probability density distribution of parametrization to produce random sample, the candidate samples that each iteration is used all changes, and therefore optimizing process is not easily absorbed in locally optimal solution. At present, Chinese scholars by this algorithm application to solving in multiple combinatorial optimization problem, and have made some progress and achievement.

Summary of the invention

The advantages such as in order to fast and reliable solution Problem of Multiprocessor Scheduling, the present invention provides a kind of method utilizing combined cross entropy to realize multiprocessor scheduling, and the method has that iteration number of times is few, stability height, working time are short.

It is an object of the invention to be achieved through the following technical solutions:

Utilize combined cross entropy to realize a method for multiprocessor scheduling, comprise the steps:

Step one: the mathematical model building Problem of Multiprocessor Scheduling, establishes objective function and constraint condition, and wherein, the mathematical model of Problem of Multiprocessor Scheduling is as follows:

$z=\underset{1\≤n\≤c}{\max }\underset{m=1}{\overset{d}{\Σ}}{y}_{nm}{t}_m$

$\begin{array}{cc}s.t.& \underset{n=1}{\overset{c}{\Σ}}{y}_{nm}=1\end{array}$

y_nm=0,1 (m=1,2 ..., d; N=1,2 ..., c);

In formula, z has represented the time required for d item operation, i.e. objective function; y_nm=1 expression operation T_mAt handler H_nUpper process;Represent handler H_nThe time of completion; t_mRepresent that handler completes operation T_mTime; HandlerRepresent operation T_mA handler completes, i.e. constraint condition;

Step 2: utilize combined Cross-Entropy Algorithm, objective function and the constraint condition established by Problem of Multiprocessor Scheduling by Matlab are programmed, are solved, and obtain objective function optimum solution.

Tool of the present invention has the following advantages:

1, the present invention is according to the constraint relation of handler and operation, multiprocessor scheduling problem is represented the linear 0-1 integer programming problem for making the minimization of object function, adopt combined Cross-Entropy Algorithm that this problem is asked optimum solution, and provide the concrete steps of combined Cross-Entropy Algorithm. Combined Cross-Entropy Algorithm is adopted to be tested by the concrete example of multiprocessor problem, and by the test result comparative analysis with simulated annealing and ant group algorithm, draw combined Cross-Entropy Algorithm that the present invention adopts draw optimum solution have that iteration number of times is few simultaneously, the advantage such as stability height, working time are short, further demonstrate the validity of this algorithm in process multiprocessor problem and practicality.

2, combined Cross-Entropy Algorithm is applied in Problem of Multiprocessor Scheduling by the present invention, by the introduction to this algorithm and Problem of Multiprocessor Scheduling, modeling and simulation, and with other heuritic approach comparative analysiss, show that this algorithm is to solving the validity of Problem of Multiprocessor Scheduling and accuracy.

3, the present invention provides a kind of brand-new thinking for the solution that can be converted into similar multiprocessor problem model.

Accompanying drawing explanation

Fig. 1 is the computation process of CCE algorithm;

Fig. 2 is the computation process of simulated annealing;

Fig. 3 is the computation process of ant group algorithm.

Embodiment

Below in conjunction with accompanying drawing, the technical scheme of the present invention is further described; but it is not limited thereto; every technical solution of the present invention modified or equivalent replaces, and not departing from the spirit and scope of technical solution of the present invention, all should be encompassed in protection scope of the present invention.

The present invention provides a kind of method utilizing combined cross entropy to realize multiprocessor scheduling, and concrete implementation step is as follows:

1, Problem of Multiprocessor Scheduling model

Problem of Multiprocessor Scheduling refers to the handler H that c platform is identical₁, H₂..., H_c, the operation T that d item is separate₁, T₂..., T_d, operation works in the way of uncorrelated mutually, and operation can work on what handler in office arbitrarily, but the operation not completed does not allow to interrupt. The separate operation of d item can not be split as less subjob. The object of scheduling provides a kind of rationally superior scheduling scheme, makes c platform handler so that the short time completes d item operation as far as possible. The target of Problem of Multiprocessor Scheduling refers under the prerequisite meeting certain performance index and constraint condition, task that can be parallel is determined a kind of assignment by suitable distribution strategy and performs order, it is reasonably allocated on each handler to perform in order, to reach the target reducing total execution time. Therefore the mathematical model of Problem of Multiprocessor Scheduling is such as formula shown in (1):

$z=\underset{1\≤n\≤c}{\max }\underset{m=1}{\overset{d}{\Σ}}{y}_{nm}{t}_m$

$\begin{array}{cc}s.t.& \underset{n=1}{\overset{c}{\Σ}}{y}_{nm}=1\end{array}$

y_nm=0,1 (m=1,2 ..., d; N=1,2 ..., c) (1).

In formula, z has represented the time required for d item operation; y_nm=1 expression operation T_mAt handler H_nUpper process;Represent handler H_nThe time of completion; t_mRepresent that handler completes operation T_mTime; HandlerRepresent operation T_mCompleting on a handler, namely the object of Problem of Multiprocessor Scheduling seeks the minimum value of z.

Feature and the known Problem of Multiprocessor Scheduling of model according to Problem of Multiprocessor Scheduling belong to discrete optimization problem, and multiprocessor problem is optimized and solves by the present invention application CE, is namely obtained the minimum value of z by Cross-Entropy Algorithm. Because y_nmValue be 0 or 1, so Problem of Multiprocessor Scheduling is 0-1 integer programming problem.

2, Cross-Entropy Algorithm

CE algorithm is a kind of overall situation random optimization algorithm proposed on the basis of information theory by Rubinstein professor the earliest, in the last few years Cross-Entropy Algorithm be applied to that trouble diagnosis, prediction etc. are large complicated, in optimization problem. The essential characteristic of this algorithm is that the candidate samples that each iteration is used all changes according to parametrization probability density distribution in optimizing process. Therefore in CE algorithm optimizing process it is crucial that iteration, the concrete process that realizes can be divided into two steps:

(1) one group of random sample is generated by given probability density function;

(2) according to the random sample update probability density function produced, and then it is next step iteration more excellent sampled data of generation.

The ultimate principle of 2.1 Cross-Entropy Algorithms

For optimization problem:

$S\left(x^{*}\right)={\mathrm{\γ}}^{*}=\underset{x\∈\mathrm{\χ}}{\min }S\left(x\right)---\left(2\right),$

In formula, S take �� as the real-valued function of field of definition; ��^*For the minimum value of S; x^*For optimum solution; �� is finite set. If probability density function race be f (; U), u �� U}, u are the parameter of density function, for given probability density function f (; V), v �� U, v are the parameter of given probability density function, and formula (2) can be converted into:

$l\left(\mathrm{\γ}\right)=P_v\left(S\right(X)\≤\mathrm{\γ})=\underset{x\∈\mathrm{\χ}}{\Σ}{I}_{\{S\left(X\right)\≤\mathrm{\γ}\}}f(x;v)=E_v{I}_{\{S\left(X\right)\≤\mathrm{\γ}\}}---\left(3\right).$

In formula, �� represents given real number; L represents the probability that S (X) is less than given real number ��; I_{{S(X)�ܦ�}}For indicator function set; X be by f (; V) random sample produced; E_vRepresent corresponding expected value.

When �� is close to ��^*Time, l value will be more and more less, and therefore in order to meaningful, l value can not be too little, then �� and v choose most important. In order to head it off, adopt multi-level algorithm, it is to construct distribution parameter sequence { u_t, t > 0} and level sequences in order { ��_t, t > 0} (t is iteration number of times). Then by v_tAnd ��_tUpgrade iteration, until the maximum value of corresponding element knots modification is less than the parameter b of a certain regulation in distribution parameter sequence after certain iteration_tol, iteration terminates.

2.2 combined Cross-Entropy Algorithms

CE algorithm is divided into successive type Cross-Entropy Algorithm and combined Cross-Entropy Algorithm (CCE), and difference between the two is the selection of probability density function, and the probability density function for the combined Cross-Entropy Algorithm of combinatorial optimization problem is Berboulli distribution. If the probability of success is p, then the probability density function of CCE algorithm is:

F (x; P)=p^x(1-p)^1-x(4)��

(4) in formula, as x=1, f (x; P)=p; As x=0, f (x; P)=1-p.

The step of CCE algorithm is:

The first step: set probability vector initial value as p₍₀₎(k is p₍₀₎Dimension), fractile coefficient ��, sample number M, smoothing factor ��, iteration number of times t=0, terminate parameter b_tol��

2nd step: make t=t+1, p_t-1The sample matrix X of M �� k is produced with Bernoulli distribution_t=[x_1(t),x_2(t),...,x_M(t)]^T, wherein x_a(t)=(x_a(t),1,x_a(t),2,...x_a(t),k), 1��a��M, M is the number of random sample; K is the dimension of each random sample vector.

3rd step: obtain objective function matrix S_(t)=[S_1(t),S_2(t),...,S_M(t)]^T, by S_(t)Carry out sorting (ascending order or fall sequence), and the matrix after sequence is designated asAnd calculate(1-��) fractile, shown in (5):

��_(t)=S_{[(1-��)M]}(5);

In formula, ��_tFor(1-��) fractile; S is objective function.

4th step: utilize (6) formula undated parameter p=(p₁,...,p_k);

$p_j=\frac{\underset{i=1}{\overset{M}{\Σ}}{I}_{\{S\left(X_i\right)\≤\mathrm{\γ}\}}{X}_{ij}}{\underset{i=1}{\overset{M}{\Σ}}{I}_{\{S\left(X_i\right)\≤\mathrm{\γ}\}}},(j=1,2,\mathrm{...}k)---\left(6\right);$

In formula, I_{{S(Xi)�ܦ�}}For indicator function set; X_ijRepresent the i-th row jth element of sample matrix; p_jFor undated parameter p jth element (j=1 ..., k).

5th step: utilize smoothing parameter ��, to p_jProcess, as shown in (7) formula;

p_j(t)=�� p_j(t)+(1-��)p_j(t-1)(7);

In formula, p_j(t)Be jth the element in argument sequence after the t time iteration (j=1 ..., k).

6th step: if the parameter matrix that adjacent twice iteration produces meets (8) formula, stop iteration, otherwise from the 2nd step iteration again.

max(|p_j(t)-p_j(t-1)|) < b_tol(8)��

Program execution exports optimum solution after terminatingOptimum value ��^*=S (X^*)=z_min��

3, sample calculation analysis

The handler that c=3 platform is identical and d=9 item operation, each operation needs the time t run_m=(81,40,26,4,65,98,53,71,15), solve below and how to make 9 operations within the time short as far as possible by 3 np complete problems that handler completes. If Y_nmFor operation T_mIt is assigned to handler H_nUpper all processing schemes, then the mathematical model of Problem of Multiprocessor Scheduling can represent again for formula (9):

$\begin{array}{c}z=\max \{Y_{nm}\&times;t_m^T\}\\ =max\left\{\left[\begin{array}{cccc}{y}_{11}& y_{12}& ...& y_{19}\\ y_{21}& y_{22}& ...& y_{29}\\ y_{31}& y_{32}& ...& y_{39}\end{array}\right]\&times;{\left[\begin{array}{cccc}{t}_1& t_2& \mathrm{...}& t_9\end{array}\right]}^T\right\}\end{array}$

$\begin{array}{cc}s.t.& \underset{n=1}{\overset{c}{\&Sigma;}}{y}_{nm}=1\end{array}$

y_nm=0,1 (m=1,2 ..., 9; N=1,2,3) (9).

The CCE algorithm of 3.1 Problem of Multiprocessor Schedulings

It is the variable y in z, z that mathematical model according to Problem of Multiprocessor Scheduling can obtain the objective function (shown in formula (9)) of CCE algorithm_nmValue be 0 and 1, therefore y_nmObeying Bernoulli distribution, therefore this algorithm can solve Problem of Multiprocessor Scheduling. In order to obtain the minimum value of objective function, utilize CCE algorithm optimization object function z, choose and be initially p₍₀₎=(0.5,0.5 ..., 0.5) (p₀Dimension be k), ��=0.85, M=60, ��=0.9, b_tol=1.0e-4. This problem being programmed with Matlab, program runs 10 times, and result is as shown in table 1.

Table 1CCE algorithm 10 operation results

Number of times	Working time (second)	Obtain the iteration number of times of optimum solution	Optimum solution
				1	0.084718	12	162
2	0.084875	13	163
				3	0.074149	11	158
4	0.079863	12	152
				5	0.085306	14	151
6	0.082122	12	158
				7	0.089665	6	163
8	0.047069	6	152
				9	0.085245	8	162
10	0.079440	10	169

Can obtaining by table 1, calculation result on average the 10th convergence, is chosen in 10 calculation result and is preferably separated, and the relation of objective function and iteration number of times is as shown in Figure 1.

CCE algorithm has searched out minimum value and the optimum value of objective function when the 7th time as shown in Figure 1, z=151, and now corresponding optimum solution is $y=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 0& 1& 1& 0& 0\\ 1& 1& 1& 1& 0& 0& 0& 0& 0\end{array}\right],$ Corresponding scheduling scheme is H₁(65,71,15), H₂(98,53), H₃(81,40,26,4), the scheduling time is 151.

3.2 method comparison analyses

The present invention is directed to Problem of Multiprocessor Scheduling adopts simulated annealing and ant group algorithm and CCE algorithm to be analyzed. Starting temperature 20000 degree in simulated annealing, final temperature 1 degree, annealing speed ��=0.95, the test result preferably separated is as shown in Figure 2; Pheromone Dauer property coefficient ��=0.8 in ant group algorithm, pheromone total amount Q=100, the test result preferably separated is as shown in Figure 3.

Fig. 2 and Fig. 3 and Fig. 1 is compared three kinds of algorithms can obtain optimum solution 151, but the performance of each algorithm has bigger gap. Table 2 is obtained by this problem being carried out repeatedly test with simulated annealing and ant group algorithm.

Table 2 three kinds of arithmetic result compare

It is short that the test result of Problem of Multiprocessor Scheduling is shown that CCE algorithm has working time than simulated annealing and ant group algorithm by comprehensive above three kinds of algorithms, fast convergence rate, the advantage that stability is high.

Claims (3)

1. one kind utilize combined cross entropy realize multiprocessor scheduling method, it is characterised in that described method steps is as follows:

Step one: the mathematical model building Problem of Multiprocessor Scheduling, establishes objective function and constraint condition;

Step 2: utilize combined Cross-Entropy Algorithm, objective function and the constraint condition established by Problem of Multiprocessor Scheduling by Matlab are programmed, are solved, and obtain objective function optimum solution.

2. the method utilizing combined cross entropy to realize multiprocessor scheduling according to claim 1, it is characterised in that the mathematical model of described Problem of Multiprocessor Scheduling is as follows:

$z=\underset{1\&le;n\&le;c}{max}\underset{m=1}{\overset{d}{\&Sigma;}}{y}_{nm}{t}_m$

$\begin{array}{cc}s.t.& \underset{n=1}{\overset{c}{\&Sigma;}}{y}_{nm}=1\end{array}$

y_nm=0,1 (m=1,2 ..., d; N=1,2 ..., c);

In formula, z has represented the time required for d item operation, i.e. objective function; y_nm=1 expression operation T_mAt handler H_nUpper process;Represent handler H_nThe time of completion; t_mRepresent that handler completes operation T_mTime; HandlerRepresent operation T_mA handler completes, i.e. constraint condition.

3. the method utilizing combined cross entropy to realize multiprocessor scheduling according to claim 1, it is characterised in that the concrete steps of described step 2 are as follows:

The first step: set probability vector initial value as p₍₀₎, fractile coefficient ��, sample number M, smoothing factor ��, iteration number of times t=0, terminates parameter b_tol;

2nd step: make t=t+1, p_t-1The sample matrix X of M �� k is produced with Bernoulli distribution_t=[x_1(t),x_2(t),...,x_M(t)]^T, wherein x_a(t)=(x_a(t),1,x_a(t),2,...x_a(t),k), 1��a��M, M is the number of random sample; K is the dimension of each random sample vector;

3rd step: obtain objective function matrix S_(t)=[S_1(t),S_2(t),...,S_M(t)]^T, by S_(t)Sorting, the matrix after sequence is designated asAnd calculate(1-��) fractile:

��_(t)=S_{[(1-��)M]};

In formula, ��_tFor(1-��) fractile; S is objective function;

4th step: utilize following formula undated parameter p=(p₁,...,p_k):

$p_j=\frac{\underset{i=1}{\overset{M}{\&Sigma;}}{I}_{\{S\left(X_i\right)\&le;\mathrm{\&gamma;}\}}{X}_{ij}}{\underset{i=1}{\overset{M}{\&Sigma;}}{I}_{\{S\left(X_i\right)\&le;\mathrm{\&gamma;}\}}},(j=1,2,...k);$

In formula,For indicator function set; X_ijRepresent the i-th row jth element of sample matrix; p_jFor jth the element of undated parameter p;

5th step: utilize smoothing parameter ��, to p_jProcess:

p_j(t)=�� p_j(t)+(1-��)p_j(t-1);

In formula, p_j(t)It is jth the element in argument sequence after the t time iteration, j=1 ..., k;

6th step: if the parameter matrix that adjacent twice iteration produces meets following formula, stop iteration, otherwise from the 2nd step iteration again;

max(|p_j(t)-p_j(t-1)|) < b_tol;

Program execution exports optimum solution X after terminating^*, optimum value ��^*=S (X^*)=z_min��